Necessary and Sufficient Condition for Nonzero Quantum Discord

Quantum discord characterizes “nonclassicality” of correlations in quantum mechanics. It has been proposed as the key resource present in certain quantum communication tasks and quantum computational models without containing much entanglement. We obtain a necessary and sufficient condition for the existence of nonzero quantum discord for any dimensional bipartite states. This condition is easily experimentally implementable. Based on this, we propose a geometrical way of quantifying quantum discord. For two qubits this results in a closed form of expression for discord. We apply our results to the model of deterministic quantum computation with one qubit, showing that quantum discord is unlikely to be the reason behind its speedup.

Figure 1

The set of two-qubit states with maximally mixed marginals (i.e., the reduced states of individual qubits are completely mixed). Physical states belong to the tetrahedron, among which separable ones are confined to the octahedron. The zero-discord states are labeled by the solid gray (red) lines (almost all states have nonzero discord [12]). The states with maximal value of discord correspond to the vertices of the tetrahedron (the four Bell states). Among the set of separable states, those which maximize discord are the centers of octahedron facets $(\pm 1,\pm 1,\pm 1)/3$ (black dots).

Figure 2

The quantum circuit for estimating the normalized trace of the unitary matrix $U_n$ using the model of deterministic computing with one quantum bit (DQC1). $H$ stands for the Hadamard gate. The control (top) qubit is measured in the ${\sigma }_1$ and ${\sigma }_2$ basis, and the expectation values give the real and imaginary parts of normalized trace $\tau =\mathrm{Tr}{U}_n/2^n$ with the overhead scaling as $\frac{1}{{\alpha }^2}$ [10].